Allen Paul

Allen Paul, Neill Campbell, Tony Shardlow

We derive an algorithm for compression of the currents and varifolds representations of shapes, using ridge leverage score (RLS) sampling, and the theory of Nystrom approximation in Reproducing Kernel Hilbert Spaces. Our method is faster than existing compression techniques and comes with theoretical guarantees on the rate of decay of the compression error as a function of the smoothness of the associated shape representation. The obtained compressions are shown to be useful for accelerating downstream tasks such as nonlinear shape registration in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework without loss of quality, even for very high compression ratios. The performance of our algorithm is demonstrated on large-scale shape data from modern geometry processing datasets, and is shown to be fast and scalable with rapid error decay.

Allen Paul, Neill Campbell, Tony Shardlow

We extend our work for compression of currents and varifolds to a compression algorithm for the embedded normal cycles representation of shape, restricted to the constant normal kernel case, using the Nystrom approximation in Reproducing Kernel Hilbert Spaces (RKHS) and ridge leverage score (RLS) sampling. Our method comes with theoretical guarantees on the compression error decay, and the approximations are shown to be effective for downstream tasks such as nonlinear shape registration in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, even for very high compression ratios. The performance of our algorithm is demonstrated on large-scale shape data from modern geometry processing datasets and is shown to accelerate downstream registration tasks significantly.

Adam Hartshorne, Allen Paul, Tony Shardlow et al.

This work presents a unified framework for the unsupervised prediction of physically plausible interpolations between two 3D articulated shapes and the automatic estimation of dense correspondence between them. Interpolation is modelled as a diffeomorphic transformation using a smooth, time-varying flow field governed by Neural Ordinary Differential Equations (ODEs). This ensures topological consistency and non-intersecting trajectories while accommodating hard constraints, such as volume preservation, and soft constraints, \eg physical priors. Correspondence is recovered using an efficient Varifold formulation, that is effective on high-fidelity surfaces with differing parameterisations. By providing a simple skeleton for the source shape only, we impose physically motivated constraints on the deformation field and resolve symmetric ambiguities. This is achieved without relying on skinning weights or any prior knowledge of the skeleton's target pose configuration. Qualitative and quantitative results demonstrate competitive or superior performance over existing state-of-the-art approaches in both shape correspondence and interpolation tasks across standard datasets.

Allen Paul, George Grammatopoulos, Adwaye Rambojun et al.

Dysplasia is a recognised risk factor for osteoarthritis (OA) of the hip, early diagnosis of dysplasia is important to provide opportunities for surgical interventions aimed at reducing the risk of hip OA. We have developed a pipeline for semi-automated classification of dysplasia using volumetric CT scans of patients' hips and a minimal set of clinically annotated landmarks, combining the framework of the Gaussian Process Latent Variable Model with diffeomorphism to create a statistical shape model, which we termed the Gaussian Process Diffeomorphic Statistical Shape Model (GPDSSM). We used 192 CT scans, 100 for model training and 92 for testing. The GPDSSM effectively distinguishes dysplastic samples from controls while also highlighting regions of the underlying surface that show dysplastic variations. As well as improving classification accuracy compared to angle-based methods (AUC 96.2% vs 91.2%), the GPDSSM can save time for clinicians by removing the need to manually measure angles and interpreting 2D scans for possible markers of dysplasia.